John F. Nash

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Past Awards

John F. Nash and Carlton E. Lemke have been awarded the 1978 John von Neumann Theory Prize for their outstanding contributions to the theory of games.

The citation reads as follows:

The theory of games was John von Neumann's most distinctive historical contribution (1928- 1944) to the fields now known as Operations Research and Management Science, so it is perhaps fitting that this year's von Neumann prize should be shared by the two major contributors to the main extension of von Neumann's original design, the theory of noncooperative games.

Further, the distinctive and complementary contributions of John Forbes Nash, Jr. and Carlton Edward Lemke to this theory have impacted areas of Operations Research and Management Science and other sciences, far beyond the formal limits of game theory. And it is particularly for these contributions among their scientific work that the prize is jointly awarded.

The earliest proofs of von Neumann's celebrated minimax theorem depended on topological arguments (1928, 1937, 1941), but the prevailing trend among mathematicians for many years was to search for "elementary" (i.e., algebraic) proofs and extensions of that result, and, indeed, to treat all of game theory as little more than a branch of the theory of linear inequalities, which was then in an exciting period of rapid growth. John Nash, however, adopted a radically different viewpoint. As a young graduate student at Princeton, he conceived the idea of noncooperative equilibrium in multi-person games and went on to prove a general existence theorem for this solution concept. His proofs (1950, 1951) are beautiful applications of the topological fixed point theorems of Brouwer and Kakutani. Indeed, their present-day familiarity among economists and others can be traced back to Nash's work, since that work was the acknowledged basis for the seminal papers of Debreu and Arrow (1952, 1954) on general equilibrium that touched off the remarkable revitalization and mathematical deepening that transformed economic theory in the 1950's and 1960's. This revolution was undoubtedly going to occur in any case, but in the actual train of events a most decisive role was played by Nash's clear penetration into the heart of a fundamental process of social interaction.

Nash's equilibrium proofs were non-constructive, and for many years it seemed that the nonlinearity of the problem would prevent the actual numerical solution of any but the simplest noncooperative games. The breakthrough came in 1964 with an ingenious algorithm for the bimatrix case (i.e., finite, two-player games) devised by Carlton Lemke and J. T. Howson, Jr. It provided both a constructive existence proof and a practical means of calculation.

The underlying logic, involving motions on the edges of an appropriate polyhedron, was simple and elegant yet conceptually daring in an epoch when such motions were typically contemplated in the context of linear programming. Lemke took the lead in exploiting the many ramifications and applications of this procedure, which range from the very basic linear complementarity problem of mathematical programming to the problem of calculating fixed points of continuous, nonlinear mappings arising in various contexts. A new chapter in the theory and practice of mathematical programming was thereby opened which quickly became a very active and well-populated area of research. Nor was the game-theory aspect neglected: the path-following methodology has been a source of many new insights into the nature of the Nash Equilibria."